Embed Size (px)
Citation preview
SIAM J. COMPUT.Vol. 20, No. 5, pp. 865-877, October 1991
1991 Society for Industrial and Applied Mathematics004
PP IS AS HARD AS THE POLYNOMIAL-TIME HIERARCHY*
SEINOSUKE TODAY"
Abstract. In this paper, two interesting complexity classes, PP and P, are compared with PH, thepolynomial-time hierarchy. It is shown that every set in PH is polynomial-time Turing reducible to a set inPP, and PH is included in BP. 0)P. As a consequence of the results, it follows that PP PH (or 03P___ PH)implies a collapse of PH. A stronger result is also shown: every set in PP(PH) is polynomial-time Turingreducible to a set in PP.
Key words, polynomial-time hierarchy, probabilistic Turing machine, polynomial-time Turing reduc-tions, parity, randomized reduction
AMS(MOS) subject classifications. 68Q15, 03D15
1. Introduction. Since the notion of probabilistic Turing machines was introducedby Gill [5], much attention has been given to several questions about its computationalpower. One ofthose questions is whether PP is more powerful than PH (the polynomial-time hierarchy), where PP denotes the class of sets accepted by polynomial-time-bounded probabilistic Turing machines with two-sided unbounded error probability.In particular, it is important in the theory of computational complexity to ask whetherPH is included in PP, or to ask whether all sets in PH are reducible to sets in PP undera suitable reducibility. This has been an open question discussed in many papers [1],[2], [10], [12], [15], [16], [19-21]. It was shown by Gill [5] that NPU co-NP is includedin PP. It is not known, however, whether A2P is included in PP. For this question,Beigel, Hemachandra, and Wechsung [3] have recently shown that pNP{ogJ is includedin PP. This is the strongest result known currently for the containment question of PHin PP. Some related results have been shown in [20].
In this paper, we give an affirmative answer to one of the above questions. Weshow that all sets in PH are _< -reducible to a set in PP. Namely, our Main Theoremin this paper is stated as follows.
MAIN THEOREM. PH P(PP)As an immediate consequence, we see that PP is not included in PH unless PH
collapses to a finite level. This gives us evidence that PP is harder than PH. In theprocess of proving the Main Theorem, we show an interesting result about the hardnessof the class P. This class was introduced by Papadimitriou and Zachos [13] andfurther investigated in several papers [13], [25], [15]. We show that all sets in PH arereducible to a set in this class under polynomial-time randomized reductions withtwo-sided bounded error probability. It was shown by Valiant and Vazirani [25] thatall sets in NP are reducible to a set in P under polynomial-time randomized reductionswith one-sided bounded error probability. These randomized reductions are stronger,but in other respects our result extends theirs. In fact, they asked how computationallydifficult P is. Our result is an answer to their open question.
Our proof of the main theorem proceeds as follows. In 3, we show that PH isincluded in BP. P, where BP. denotes the BP-operator introduced by Sch/Sning 15].Intuitively speaking, a set is in BP. P if and only if it is reducible to a set in P
* Received by the editors February 21, 1989; accepted for publication (in revised form) December 12,1990. This research was supported in part by International Information Science Foundation grant 89 2 2 176.
t Department of Computer Science and Information Mathematics, University of Electro-communica-tions, 1-5-1 Chofugaoka, Chofu-shi, Tokyo 182, Japan.
865
866 SEINOSUKE TODA
under a polynomial-time randomized reduction with two-sided bounded error probabil-ity. The proof of it is based on a result by Valiant and Vazirani [25] and a result bySch/Aning [15]. In 4, we show that BP. 0)P is included in P(PP). In fact, we willshow a stronger result than this. The proof is based on a structural property of 0)Pdiscovered in this paper. At the end of 4, we will mention a stronger result than theMain Theorem above: PP(PH) P(PP). This result is obtained by combining thetechnique in this paper with a result by K6bler et al. [10].
2. Preliminaries. We assume that the reader is familiar with the basic conceptsof computational complexity theory. Let E be a finite alphabet. For a string w E*, Iwldenotes the length of w. For a set L
_E*, L denotes the complement of L. For a class
K of sets, co-K denotes the class of sets whose complement is in K. Let En (respectively,E -<n and E<’) denote the set of strings with length n (respectively, length, at most nand less than n). For a finite set X
_E*, IIXII denotes the number of strings in X. Let
N denote the set of natural numbers.Our sets in this paper are over E {0, 1, : } unless otherwise specified. The symbol
: is usually used as a delimiter among strings of {0, 1}*. A pairing function (respectively,a k-tuple function) over {0, 1}* is represented by delimiting two strings (respectively,k strings) by this symbol.
Our models of computation are variations of polynomial-time-bounded oracleTuring machines (deterministic, nondeterministic, or probabilistic). Our oraclemachines are usual ones. For an oracle machine M and an oracle set A, M(A) denotesthat M uses A as an oracle. A polynomial-time-bounded deterministic (respectively,nondeterministic) oracle machine is abbreviated by an oracle P-machine (respectively,oracle NP-machine ). A polynomial-time-bounded probabilistic oracle machine withtwo-sided unbounded error probability (respectively, with two-sided bounded errorprobability) is abbreviated by an oracle PP-machine (respectively, an oracle BPP-machine). In the unrelativized cases, we omit the term "oracle." For example, an oracleNP-machine with the empty set as an oracle is simply called an NP-machine.
For an oracle set A, P(A) denotes the class of sets accepted by oracle P-machinewith oracle A. NP(A), PP(A), and BPP(A) are defined similarly. 0)P(A) denotes theclass of sets L for which there exists an oracle NP-machine M such that for each x, xis in L if and only if the number of accepting computation paths of M(A) on x is odd.This class was defined by Papadimitriou and Zachos [13]. For a class K of oracle sets,P(K) {P(A): A K}. Other classes are defined similarly. The unrelativized classesare defined by setting the oracle set to the empty set, and the specification of oracleset is omitted in this case.
We assume that all polynomial-time-bounded oracle machines M satisfy thefollowing conditions.
(1) Its transition function has at most two possible transitions from eachconfiguration.
(2) All computation paths of M are encoded into a string of {0, 1}* by the usualmanner, where a computation path may contain possible answers from a given oracle,and the oracle answer "yes" (respectively, "no") is encoded by 0 (respectively, 1).
These assumptions are technical ones. Obviously, we lose no generality underthese assumptions.
Let X be a finite set of strings and R be a predicate over strings. In this paper,we denote by Prob ({we X" R(w)}) the probability that R(w) is true for randomlychosen w from X under uniform distribution. In [15], Sch/Sning introduced theBP-operator, which produces a probabilistic class from a given class. He also defined
PP IS AS HARD AS PH 867
-operator in [16], as an abstraction of the class P. In [26], Wagner defined thecounting operator, based on a characterization of PP in [14], [26]. We give thosedefinitions here. The following definition of the counting operator is different fromthe original one; however, it is easy to see that both definitions define the same concept.
DEFINITION 2.1 [15], [16], [26]. Let K be a class of sets and let L be a set. Thenwe define some new classes denoted by @ K, BP. K, and C. K as follows.
(1) L e @ K if there exist a set A e K and a polynomial p such that for all x e E*,
x e L ---> [[{w e {0, 1}P(lxl): x 4 w e a}][ is odd.
(2) L e BP K if there exist a set A e K, a polynomial p, and a constant a > 0 suchthat, for all x e E*,
Prob ({we{0, 1}p(Ixl’xweA<--xe L})>=1/2a.
(3) L e C K if there exist a set A e K and a polynomial p such that for all x e E*,
Prob ({we{0, 1}p(lxl’xqeweA<-->xe L}) >1/2.It is easy to see that C-P= PP, BP. P= BPP, and @. P= @P. The following
propositions are basic properties of the above operators, which follow immediatelyfrom the definitions. These will be used implicitly in later arguments.
PROPOSITION 2.2. Let K1 and K2 be arbitrary classes of sets. Then, the followingstatements hold.
(1) If K_K2, then K
_@) K2.
(2) If K is closed under marked union with sets of theform {x#0 pII>" x eX*} (forarbitrary polynomial p such sets are in P), then K1 is closed under complernentation.
(3) If K1 is such that, for all sets Le K, the set {xx" x e L} also belongs to K1,then K @ K.
PROPOSITION 2.3. Let K and K2 be any classes ofsets. Then thefollowing statementshold.
(1) If KI_ K2, then BP. KI BP. K2.(2) co-BP. K
_BP. co-K Hence, if K is closed under cornplernentation, then
co-BP. KI= BP. K1.(3) If K1 is closed under padding i.e., L e K1 implies {x 44= y" x e L and y e {0, 1 }*} e
K for each set L), then K BP" K.PROPOSITION 2.4. Let Ki and K2 be arbitrary classes of sets. Then the following
statements hold.(1) If K _K, then C.K_C.K2.(2) co-C. K
_C. co-K1. Hence, if K1 is closed under complernentation, then co-
C. K1-C. K.(3) If K is closed under padding, then K1 C K(4) BP" K
___C. K
We can easily see that all the classes to be built in this paper satisfy the. closureproperties mentioned in the above propositions (except possibly for complementation).For example, II(k => 0) is closed under taking marked union with the set of the form{x#O p(lxl)"
X eY-,*}’, hence, . II is closed under complementation; we will use thisfact in the next section.
In the later sections, we will be concerned with several reducibility notions definedbelow.
DEFINITION 2.5. Let A and B be arbitrary sets. A is said to be -< P.-reducible toB(A<-P,, B) if there exists a function f computable in polynomial time such that for
P P B) ifeach x, x e A if and only ill(x)e B. A is said to be <- ma- reducible to B(A --<ma
868 SEINOSUKE TODA
there exists a function f computable in polynomial time such that for each x,f(x)=yl#Y2# #Ym (m >--_ 1), and x A if and only if the majority of yi’s are in B. A issaid to be <- -reducible to B (A --<TP B) if there exists an oracle P-machine that acceptsA with oracle B. Let K be a class of sets and let <-rP denote an arbitrary reducibility.A set L is <P-r-hard for K if every set in K is _< P-reducible to L; if L K in addition,then L is said to be --r-< P complete for K. Then we say that K is closed under < p if andonly if for all sets A and B, A _< P B and B If implies A K.
PObserve that for all sets A and B, A maj B implies A _< TP B; and hence, if a class
PIf is closed under <--, then the class is closed under <--maj.
3. P is hard for PH under randomized reducibility. In this section, we show that)P is hard for the polynomial-time hierarchy under polynomial-time randomizedreducibility. More precisely, our main result in this section is stated as follows.
THEOREM 3.1. PHi_ BP" P.Before proving this, we give an intuitive explanation of the proof to the reader.
We first show that E is included in BP. II_ for each k > 1 (see Lemma 3.3)This generalizes a result due to Valiant and Vazirani [25] in which they showed thatall NP-complete sets are reducible to a set in P under randomized polynomial-timereducibility. Our proof technique is essentially the same as theirs. We next observethat it is possible to swap a @-operator and a BP-operator. In particular, we showthat @ BP. @ P__G_ BP. @ @ P (see Lemma 3.6). Furthermore, we observe that it ispossible to reduce two consecutive BP-operators (respectively, two consecutiveoperators) to one operator: It was shown by Papadimitriou and Zachos [13] that0) P(@ P) @ P. This implies that 0) @ P @ P, and we also show that BP BP @ PBP 0)P (see Lemma 3.7). At the end of this section, we put all this together to proveTheorem 3.1, using an induction on the levels of the polynomial-time hierarchy.
Now we begin to show the lemmas mentioned above. Following Valiant andVazirani [25], we shall view strings of {0, 1} as n-dimensional vectors from the vectorspace GF[2] n. We denote by u. v the inner product of two vectors u and v over GF[2].In [25], they showed the following result.
THEOREM 3.2 [25]. Let n>= 1 and let So__ {0, 1} be a nonempty set. SupposeWl, w2," w, are randomly chosen from {0, 1}". Let So S and let
S={v S: v. w= v" w2 v. w=O}
for each 1 <= <-_ n. Let P,(S) be the probability that Si 1 for some 0 <- n. Then,P,(S) >=-.
LEMMA 3.3 For each k > 1, Z U 1-I c B P. . 1-I_ |o
Proof By Propositions 2.2(2) and 2.3(2), BP .. 1-I_ is closed under com-plementation. Hence, it suffices to show that c_ BP. . II_. Let L . Then itwas shown by Stockmeyer [18] and Wrathall [28] that there exist a set A II_ anda polynomial p such that for every x, x L if and only if x y A for some yWe define a set C as follows:
C {x :: w w2 Wp(Ixl): for each i, 1 <= <- p([xl) w {0, 1} p(lxl), and for some j,
O<-j<-p(Ixl), II{Y {0, 1} p(Ixl)" x y A^ (ti<=j)[w;. y =0]}11 is odd}.
We first show that C is in II_l. Since FI _1 is closed under complementa-tion, it suffices to show that C’s complement, , is in . II_l. This can be done asfollows: Given arbitrary strings x and z wlw2 Wpll such that, for each i, 1P(lX[), wi {0, 1}
PP IS AS HARD AS PH 869
x#zC-for each j, 0 _-<j <- p([x]), []{y {0, 1} ,(Ixl). x # y A ^ (/i <_j)[wj. y 0]}[[ is even
o n (ll{" x#yJ^(Vi<_j)[wj.y=O]}ll+l)isodd.j=0
Hence we may define a set B 6H_ and a polynomial q such that for every x # z,(Izl =p(Ixl)2),
p(lxl)
j=0
The set B is defined by
B {x # w wp(lxl) # ayay. a(ll)w(ixl)" for every j, 1 j p(]x[),
wj {0, 1 } v(ll, aj {0, 1 }, yj {0, 1 } v(ll, and(W, J p(Ixl))[ajyj o+’ v (j x yj A (Vi j)[w," yj 0])]}.
It is easy to see that BH_ and that B and the polynomial q(n)=p(n)(1 +p(n))satisfy the required condition; that is, the set B and the polynomial q witness C.Pk-l"
Next, we show L BP. . H_ by using the set C. Let x be a string and letw, w2," , w(ixl> be randomly chosen from {0, 1}P(lxl). We define
So {y {0, }v(- x y A}
and
S ={y So" w y w. y wi" y}
for each 1 ip(Ix[). Let Pv(ll(So) be the probability thatp(]x]). Then it is easy to see that
Hence, from Theorem 3.2,(1) xLProb({u{O, 1}P(ll)’x#uC})and(2) xeLProb({u{0,1}v(ll’x#uC})=0.
The probability of (2) follows from the fact that for all x, if x L, then x # y A forevery y {0, 1}(Ixl). To amplify the probability in (1), we fuher define a set D asfollows"
D {x # u,uu lu, (Ixl) for each 1, 2, 3 and x # u, C for some 1, 2, 3}.
Then we obtain that for each x,
(3) xLProb({uuu{O, 1} v(ll)" x#uuuD})
=1 (Prob ({u {0, 1} (ll)" x # u C}))
1 27-=+ and
(4) xLProb({uuu{O, 1}v(ll)’x#uuuD})=O.By using the same argument as when showing C H_, we can show D H_.This implies L
It was shown by Papadimitriou and Zachos 13] that P( P) P. This impliesthe following theorem. For the sake of making this paper more selficontained, weprovide a sketch of their proof.
870 SEINOSUKE TODA
THEOREM 3.4 [13]. 0) P(0) P) P. Hence we have that O) P P and thatPP is closed under --<maj.
Proof Sketch. Let L be a set in 0)P(0)P). Then there exist a set A 0)P and anoracle NP-machine M such that for every x, x L if and only if the number of acceptingpaths of M(A) on input x is odd. Let M1 be an NP-machine that witnesses A P.Then we define an NP-machine M2 working on a given input x as follows"
(1) M2 first guesses a computation path w ofM on input x, which includes possibleoracle answers to the query strings appearing in w.
(2) If w is a rejecting path of M on x, then M2 enters a rejecting state; otherwise,it goes to the next step.
(3) Let y, Y2, Y,, (z, z2, , Zl) be all the query strings which appear in wand whose corresponding oracle answers in w are "yes" (respectively, "no").Then M simulates M successively for each y and each z in the followingmanner:(a) For each yi, it simply simulates M. If M enters a rejecting state, thenso does M2; otherwise, it proceeds to the next simulation.(b) For each z, it nondeterministically selects one of the following processes:
(i) M2 goes to the next simulation. (Intuitively speaking, this processcan be regarded as a dummy-accepting path of M1 on input z.)
(ii) M2 simulates M on z. If M1 enters a rejecting state, then so doesM2; otherwise, it goes to the next simulation.
(4) M2 enters an accepting state.
We can classify all possible accepting computation paths of M on input x intotwo groups, one of which consists of accepting paths of M(A) on x (group 1), andthe other consists of the remaining ones (group 2). Obviously, every accepting pathin group 1 contains correct oracle answers of the oracle set A, and every acceptingpath in group 2 contains a wrong oracle answer. From the definition of M2, we caneasily see that every accepting path in group 1 is followed by an odd number ofaccepting paths in steps 3 and 4, and every accepting path in group 2 is followed byan even number of accepting paths in those steps. From this observation, it is notdifficult to see that for every x, x L if and only if the number of accepting paths of
M on input x is odd. We leave the verification to the interested reader. Thus L is inP. The other statements are immediate from the first one.The following theorem was shown by Sch6ning [15].
PTHEOREM 3.5 [15]. Let K be a class of sets which is closed under --<maj. Then forall sets A B P K and all polynomials q, there exist a set B and a polynomial p such that,for all n,
Prob ({y {0, 1} p(n)" (X, ]X]-- n)[x y B--xa]})>= 1-2-q(n).
LEMMA 3.6. 0) BP ( P c__ BP P.Proof Let L . BP. P. Then there exist a set A BP. P and a polynomial
p such that for each x, x L if and only if
[l{w" Iwl =p(Ixl) and x w
is odd. Furthermore, there exist a set B P, a polynomial q, and a constant a > 0such that for each y,
Prob ({u {0, 1} q(lyl)" y=uB,-y6A})>--+o.PSince P is closed under --<ma, we may assume, from Theorem 3.5, that for all m,
Prob ({u {0, 1}q("): (Vy, lyl m)[y. u6 B--yA]})>-1/4.
PP IS AS HARD AS PH 871
Hence we have that for all x of length n,
Prob ({u e {0, 1} q(n+l+p(n))" (VW e {0, 1}P(n))[X # W # 12 e B *- x # w e A]}) >- ,and hence,
(1) Prob({ue{O, 1}q(n+l+p(n))’{we{O, 1}p(n)’x#w#ueB}
={we{O, 1}P(": x# w e A}}) >-.For a string x of length n, assume xe L. Then [l{w e {0, I}P(": x wea}[I is odd.
Hence from (1),
(2) Prob ({ue{O, 1} q("+l+p("))" [[{we{O, 1}P("): x# w#ueB}l is odd}) >_- .Conversely, assume xeL. Then [[{we{O, 1} p("" x# weA}l[ is even. Hence from (1),
(3) Prob ({ue{0, 1} q(n+l+p(n))" [[{We{0, 1}P(n): X# w#ueB}l is odd}) -<_ 1/4.
We now define B’ and A’ by
B’={x# u # w" lu[= q([xl+ l+p([x[)), [w[=p([x[), and x# w # u e B}
and
A’= {x * u" lul q(lxl + 1 +p(lxl)) and x, u * w e B’
for an odd number of w
Then we obtain that Prob ({u e {0, 1} q(Ixl+l+p(Ixl))"X U e A’ ’-’ x e L}) >-- from (2) and
(3) above. It is easy to see that B’e 03 P and A’e ). P. Hence, A’ is in 03 P fromTheorem 3.4. This implies L e BP. 03 P.
LEMMA 3.7. BP" BP q) P BP 0) P.Proof. It suffices to show the inclusion BP. BP. q)PG BP. 03 P. Let L e BP. BP.
0)P. Then there exist a set A e BP. O)P and a polynomial p such that for each x,
Prob ({we{0, 1}p(Ixl): x weA<->xeL})>-.
Furthermore, there exist a set B e @ P and a polynomial q such that for each y,
Prob ({u e {0, 1}q(Iyl):
Note that we are using Theorem 3.5 in this setting. We now define a set C by
C {x # wu" [w[ P(lx[), [u[ q(lx # w[)= q(lx[ + 1 +p(Ix})), and x # w # u e B}.
It is easy to see that C is in 03 P. For a string x, if x e L, then(1) Prob ({wu e {0, 1} p(IxI)+q(IxI+I+p(IxI))" x # wu e C})
=Prob ({we{0, 1} p(II" x# weA})
xProb ({u e {0, 1}q(lxl+l+p(lxl)): x# w# Ue B}[x# weA)
+Prob ({we{0, 1}p(Ixl): x# weA})
x Prob ({u e {0, 1}q(Ixl+l+p(Ixl)): X W U e B}lx w A)
872 SEINOSUKE TODA
where Prob (X/Y) denotes the conditional probability of the event X under thecondition Y.
Conversely, if x L, then(2) Prob ({W/,/C {0, 1}p(lxl)+q(lxl+l+p(lxl)): x Wbl C})
=Prob ({w {0, 1} p(Ixl)" x wA})
x Prob ({u {0, 1}q(Ixl+l+P(Ixl)): x w u B}lx w A)
+Prob ({w {0, 1} p(Ixl) x#t wf:A})
Prob ({u {0, 1}q(Ixl++p(Ixl)): X W 4 u B}lx w A)
_-<z.l+l.= =-.Thus we have that for every x,
Prob ({v {0, 1} p(lxl)/q(lxl+/p(lxl)), x v C x Z}) -> 1/2+.This implies that L
Now we can prove Theorem 3.1.Proof of Theorem 3.1. We prove this theorem by induction on the levels of the
polynomial-time hierarchy. The inclusion E P BP. P is obvious. We now assumec BP. @P for some k>0. It is easy to see that BP. @P is closed under com-
plementation. Hence we have the inclusion IIkP_l_BP. 03 P. Then,
;c__ BP. @. 1-I[_ (from Lemma 3.3)_BP. @ BP. @P (from inductive assumption)
BP. @ P (from Lemma 3.6 and Lemma 3.7).
Thus we conclude that PH U k__>0 Z[-----BP. @ P.It was shown by Sch6ning [15] that IIc_ BP. Z[ implies Z+l=H[+ for every
k _-> 1, which is regarded as a refinement ofthe result by Karp and Lipton [7]. Combiningthis result with Theorem 3.1, we observe that P is harder than PH unless PH collapsesto a finite level. More precisely, we obtain the following corollary.
COROLLARY 3.8. For every k >- 1, if P_ ,, then PH E+l. Hence 03 P_ PH
implies that PH collapses at a finite level.The second statement in this corollary follows from the fact that 03 P has a complete
set under _< Pro-reducibility.
4. PP is <-_-hard for PH. In this section, we prove the following theorem.THEOREM 4.1. C @ P
_P(PP).
It is easy to see that BP. @ P_ C. @ P. Hence the Main Theorem in 1 followsimmediately from this theorem and from Theorem 3.1.
The following lemma plays an important role in the proof of Theorem 4.1, anddepends on an interesting numerical property. For an NP-machine N and an input y,let # aCCN(y) denote the number of accepting computation paths of N on input y.
LEMMA 4.2. Let X be a set in P and let q be a polynomial. Then, there exists anNP-machine N such that for each input y of length n,
(1) if ye! X, then # accr(y)=-O.(mod 2q(),and(2) ify X, then # accrl(y) -1 (mod 2q(")).Before proving this lemma, we give an intuitive explanation about our proof of
Theorem 4.1 and about the role of Lemma 4.2. Let L be a set in C. 03 P. Then thereexist a set X @ P and a polynomial p such that for every x,
Prob ({ w c {0, 1 } p(lxl), x # W G X iff x L}) > 1/2.
PP IS AS HARD AS PH 873
Let N1 be an NP-machine satisfying the conditions in Lemma 4.2 for the set L andthe polynomial q(n) n. Consider an NP-machine N which operates as follows. Givenan input x of length n, N first guesses a string w of length p(n), and it begins tosimulate N1 on input x # w. If N1 enters an accepting state, then N enters an acceptingstate; otherwise, it enters a rejecting state. Let # X[x] denote the number of stringsw such that Iwl=p(n) and x wX. From Lemma 4.2, it is not difficult to see that# aCCN(X)=2n+l+p(n kx- X[x] for some natural number kx-->0. By a standardbinary search technique, # aCCN(X) can be computed in polynomial time with an oracleset from PP. After this, we can also get the value of #X[x] within polynomial timeby computing :aCCN(X mod 2n+l+p(n), because #X[x]<-2 p(")2n++p(n). Finally, wedecide to accept the input x if and only if #X[x] > 2 p(n)-l. Hence, we can concludethat C. P c_ p(pp).
The recurrence relation in the next lemma provides the key numerical property.Intuitively speaking, it gives us a whole exponential factor of freedom in our counting.
LEMMA 4.3. For an integer m, define the sequence So, s, s2,’’’, inductively bySo m and for > 1, si 3 4
S i- -JI- 4 s i-- 1" Then,(1) if m is even, then for all i, si is a multiple of 22, and(2) if m is odd, then for all i, si + 1 is a multiple of 2 2’.Proof. The statement (1) is obvious from the definition of the sequence. We prove
(2). The case i=O is obvious. We assume that for some i> 0, Si_ --22i-1" ki_ -1 forsome positive integer ki_. Then, from the definition of si,
S 3 4Si_ +4" Si_
3. (2 2’-1. ki-1- 1)4+4. (22i-1. k,-1-1)3
3 22i+1 4 2 3. 2 2 2‘ 2ki-1- 8. k,3._l +6. ki-1 1
22i (3 22i k,4._l 8 22’- 2ki-1 + 6" ki_ 1) 1.
Hence si is a multiple of 2 2’. [3
Accordingly, we make the following recursive definition of a functionfs" * X N-+N, where is the input alphabet of a given NTM N.
DEFINITION 4.4. For an NP-machine N and an input y, define
fN(Y, O)= aCCN(y), and for -> 1,
fN(Y, i)= 3" (fN(Y, i-1))4+4 (fN(Y, i-1))3.
LEMMA 4.5. Let N be an NP-machine, and let q( n) be a polynomial. Then for allinput y of length n,
(1) if aCCN(y) is even, thenfN(y, [log2q(n)])=-O(mod2q(")), and(2) if aCCN(y) is odd, then fN(y, [log2 q(n)])--=--1 (mod 2q(")).Proof Since 22[1g2 q(n)] 2q(n)+k 2q(n) 2k for some natural number k, this follows
immediately from Lemma 4.3.Given a set Xe @P, it follows from the definition of P that there is an
NP-machine N such that for all y, y e X if and only if aCCN(y) is odd. Hence, toobtain Lemma 4.2, it remains to show how to construct an NP-machine Q such thatfor all y,
acco(y)=fN(y, [log2 q(n)]).
This is accomplished in the following lemma.
874 SEINOSUKE TODA
LEMMA 4.6. Let N be an NP-machine, and let be a polynomial which bounds theruntime of N. Then we can find an NTM Q which takes inputs of the form y : li, anda constant c > 0 such that for all y, i:
(1) accQ(y 4 I i) =fN(Y, i), and(2) all computations of Q on input y 1 halt within c. 4i. (t(lYl) / 1) steps.Proof. Intuitively speaking, the required machine Q is designed so that for each
input y 1 , it executes itself recursively on input y : 1-1 according to the definitionof fN. This can be done by using stack operations. Furthermore, since the depth ofrecursive executions for input y 1 is at most i, and at most four sequential calls aremade at each level, the required time bound is obtained. We now describe Q as arecursive procedure, using a stack for the recursive executions.
PROCEDURE Q(y, i), where the input is written in the form y 4 1 i.Step 1: if 0 then simulate M on input y;
if M enters an accepting statethen return "ACCEPT" else return "REJECT";
Step 2: guess one of the following subprocesses nondeterministically;(subprocess 1)branch away nondeterministically into three branches;execute the following in each branch;for j:= 1 to 4 do
execute Q(y, 1) recursively;{this can be done by pushing i,j and return position into a stack beforeexecution and by popping those off the stack after execution}
if this call to Q(y, i-1) returns "REJECT" then return "REJECT"od;return "ACCEPT""(subprocess 2)branch away nondeterministically into four branches;execute the following in each branch;forj:=lto3 do
execute Q(y, 1) recursively;{this can be done by pushing i,j and return position into a stack beforeexecution and by popping those off the stack after execution}
if this call to Q(y, i-1) returns "REJECT" then return "REJECT"od;return "ACCEPT."
By induction on i, it is not difficult to show that for each input y 1 , the numberof accepting computation paths of Q is equal to fN(Y, i). The essence of this proof isto estimate the runtime of the above machine. Let y I be an input for Q and letT(y, i) denote the runtime of Q on input y 1 . It is not difficult to see that stackoperations and the other bookkeeping operations in Step 2 can be done within timeat most O(i), say c. i+ c for some c> 0, if we denote natural numbers by unarynotation. Furthermore, the operations in Step 1 can be done within a constant time,say c > 0. Then, we obtain the following inequalities from the definition of Ml:
(1) T(y, 0)<-_ t(lyl)+ c,
and
(2) T(y,i)_-<4.(T(y,i-1)+c.i+c) for each > 0.
PP IS AS HARD AS PH 875
From this, we have:
(3) T(y, i)<4 t(lY[)+ 4k =4’ 4=, "(c" i+c) t(lY[)+’(4i-1) "(c" i+c).
Thus we finally have T(y, i)<= 0(4i" (t([y[)+ i)). This completes the proof.Proof of Lemma 4.2. Let N1 on input y simulate Q of Lemma 4.6 on input
y # 1 rg2q(lYll. Then N1 runs in polynomial time in [y[, and satisfies the propertiesrequired in Lemma 4.2.
Before deducing Theorem 4.1, we state and prove a technically stronger result. Afunction h" 2" N is said to belong to the class # P [23], [24] if there is an NP-machineN such that for all x *, h(x)= #aCCN(X). Then pptl stands for the class of setswhich can be solved in polynomial time with one free evaluation of a P function.Papadimitriou and Zachos [13] showed that pNpoga pp (calling the latter class"+P"). We show that the whole polynomial-time hierarchy is contained in PPt, asa consequence of Theorem 4.7.
THEOREM 4.7. C @ P p.P[1].
Proof Let L C. @ P. Then there is a set X @ P and a polynomial p such thatfor all x, putting W {w {0, 1 } p(ll), x # w X}, we have x L if and only if Wx >2p<lxl)-l. Then from Lemma 4.2, we can find an NP-machine N such that for all inputsy, with m
(1) if y X, then for some integer ay > 0, accy(y) 2. ay 1, and(2) if y X, then for some integer/3y _-> 0, accN(y)= 2m’y.Now let Z be an NTM which on every input x of length n does this"(1) Guess w {0, 1} p(n).(2) Simulate N on input x # w, accepting if and only if N accepts.Then Z clearly runs in polynomial time. Now writing r for {0, 1} P("- W, we
have"
#accz(x) 2I,V
2VV"
#accN(x#w)+ #accN(x#w)
(21.wl.ce.w_l)+ E 21 *wl’/3w)w
Since every a and fl. is integral, it follows that # accz(x) + 11W[I is a multipleof 2II++p(II. Since wxll--<2(x <2x+/(x, it follows that wxll can be computedsimply by complementing the last p(Ixl) bits of #accz(x) in binary notation. That isto say, x L if and only if the p([xl)th bit of #accz(x) from the right is a "0." SinceZ is a polynomial-time-bounded NTM, #accz(" is a # P function, and the theoremfollows. [3
Proof of Theorem 4.1. It is well known that for every P function h, its graph{x k: h(x)<-k} belongs to PP [14]. By the standard binary search technique, h(x)can be computed with O(Ixl)-many queries to its graph. So C. ( P, and hence PH,is included in P(PP). [3
The following corollary is straightforward from the Main Theorem.COROLLARY 4.8. For each k >- O, if PP
___Z, then PH collapses to E. Furthermore,
if PP PH, then PH collapses to a finite levelProof Assume PP
_ . It is well known that PP is closed under complementation.Hence we have PP__ EfqI/ from the assumption. It is also well known that P(E1-I)_ Z for each k_-> 0. From the main theorem, we have
PH___P(PP) P(E CI II)
_ .
876 SEINOSUKE TODA
The second statement is easily obtained from the fact that PP has a complete set under-< Pm-reducibility. [3
At the end of this section, we observe a result stronger than the Main Theorem.It was shown by K6bler et al. [10] that PP(BPP)= PP. Furthermore, the equality canbe relativized to all oracle sets. More precisely, we have the following result.
THEOREM 4.9 [10]. For all oracle sets A, PP(BPP(A))= PP(A).From this theorem, we have the following theorem.THEOREM 4.10. PP(PH) is included in P(PP).Proof It is easy to see that PP(PH)PP(BP.P)_PP(BPP(P)). These
inclusions follow from Theorem 3.1 and from the definition of BP-operator. FromTheorem 4.9, we have PP(PH)_ PP(P). It is not hard to show that PP(P)=C. P(P)=C.P. Some techniques for showing this have appeared in [21], [26].Hence we obtain this theorem from Theorem 4.1.
5. Concluding remarks. In this paper, we showed that every set in PH (and inPP(PH)) is polynomial-time Turing reducible to a set in PP. We also show a similarresult about P. There are some further questions that are related to this work. Asimple question is whether we can show, by using a different kind of reducibility suchas polynomial-time truth-table reducibility, that PH is reducible to PP. In fact, weshowed that PH is included in p.Plj; this is a somewhat stronger statement thanPH G P(PP). On the other hand, it is well known that every set which is _<tPt-reducibleto a set in PP is in P*PIJ; but the converse is unknown. Hence the answer to the abovequestion will give us a somewhat stronger result than the present result. The otherinteresting question is whether C=P [26], [21] is as hard as PH. A more importantquestion is whether NP(PP) is included in P(PP), or whether PP(PP) is included inP(PP). It is also interesting to find oracle sets that separate those classes from eachother.
Acknowledgment. I am very thankful to Osamu Watanabe for helpful discussionsand some nice advice, to Lane Hemachandra for his comments on this work, and to
Kenneth Regan for his many suggestions on the earlier version of this paper. I amvery thankful to the Program Committee of the 4th IEEE Conference on Structure in
Complexity Theory in 1989 for giving me an opportunity to talk about this result at
the conference; concerning this, I have to thank Richard Beigel, Lane Hemachandra,and Gerd Wechsung for giving much of their lecture time to me. I want to thank thereferees of this paper. In particular, one of the referees gave me many suggestionswhich made the quality of this paper much better. I would also like to thank RonBook and Janos Simon for their suggestions.
REFERENCES
D. ANGLUIN, On counting problems and the polynomial-time hierarchy, Theoret. Comput. Sci., 12 (1980),pp. 161-173.
[2] J. L. BALC,ZAR, R. g. BOOK, AND U. SCHONING, The polynomial-time hierarchy and sparse oracles,J. Assoc. Comput. Mach., 33 (1986), pp. 603-617.
[3] R. BEIGEL, L. A. HEMACHANDRA, AND G. WECHSUNG, On the power ofprobabilistic polynomial-time:pNP[log] pp, in Proc. 4th IEEE Conference on Structure in Complexity Theory, 1989, pp. 225-227.
[4] JIN-YI CAI AND L. A. HEMACHANDRA, Or/the power ofparity polynomial time, in Proc. Symposiumon Theoretical Aspects of Computer Science, Lecture Notes in Computer Science 349, 1989,Springer-Verlag, Berlin, pp. 229-240.
[5] J. GILL, Computational complexity of probabilistic Turing machines, SIAM J. Comput., 6 (1977),pp. 675-695.
PP IS AS HARD AS PH 877
[6] L. A. HEMACHANDRA, On ranking, in Proc. 2nd IEEE Conference on Structure in Complexity Theory,1987, pp. 103-117.
[7] R. KARP AND R. LIPTON, Some connections between nonuniform and uniform complexity classes, inProc. 12th ACM Symposium on Theory of Computing, 1980, pp. 302-309.
[8] K. Ko, Some observations on the probabilistic algorithms and NP-hard problems, Inform. Process. Lett.,14 (1982), pp. 39-43.
[9] J. KtBLER, U. SCHtNING, AND J. TORAN, On counting and approximation, Acta Inform., 26 (1989),pp. 363-379.
[10] J. KOBLER, U. SCH(SNING, S. TODA, AND J. TORAN, Turing machines with few accepting computationsand low sets for PP, in Proc. 4th IEEE Conference on Structure in Complexity Theory, 1989,pp. 208-215.
[11] C. LAUTEMAN, BPP and the polynomial-time hierarchy, Inform. Process. Lett., 14 (1983), pp. 215-217.
[12] T. J. LONG AND A. L. SELMAN, Relativizing complexity classes with sparse oracles, J. Assoc. Comput.Mach., 33 (1986), pp. 618-627.
13] C. H. PAPADIMITRIOU AND S. ZACHOS, Two remarks on the power ofcounting, in Proc. 6th Gesellschaftfiir Informatik Conference on Theoretical Computer Science, Lecture Notes in Computer Science
145, 1983, Springer-Verlag, Berlin, pp. 269-276.[14] J. SIMON, On the difference between one and many, in Proc. 4th Colloquium on Automata, Languages
and Programming, Lecture Notes in Computer Science 52, 1977, Springer-Verlag, Berlin,pp. 480-491.
15] U. SCH)NING, Probabilistic complexity classes and lowness, in Proc. 2nd IEEE Conference on Structurein Complexity Theory, 1987, pp. 2-8; also in J. Comput. System Sci., 39 (1989), pp. 84-100.
16] , The power ofcounting, in Proc. 3rd IEEE Conference on Structure in Complexity Theory, 1988,pp. 2-9.
[17] M. SIPSER, A complexity theoretic approach to randomness, in Proc. 15th ACM Symposium on Theoryof Computing, 1983, pp. 330-335.
[18] L. J. STOCKMEYER, The polynomial-time hierarchy, Theoret. Comput. Sci., 3 (1977), pp. 1-22.[19] ,On approximation algorithms for 4P, SIAM J. Comput., 14 (1985), pp. 849-861.
[20] S. TODA, Restricted relativizations ofprobabilisticpolynomial-time, Theoret. Comput. Sci., 1990, accepted.[21] J. TORAN, An oracle characterization of the counting hierarchy, in Proc. 3rd IEEE Conference on
Structure in Complexity Theory, 1988, pp. 213-223.[22] L.G. VALIANT, Relative complexity ofchecking and evaluating, Inform. Process. Lett., 5 (1976), pp. 20-23.[23] ., The complexity of computing the permanent, Theoret. Comput. Sci., 8 (1979), pp. 189-201.
[24] The complexity ofreliability and enumeration problems, SIAM J. Comput., 8 (1979), pp. 410-421.
[25] L. G. VALIANT AND V. V. VAZIRANI, NP is as easy as detecting unique solutions, Theoret. Comput.Sci., 47 (1986), pp. 85-93.
[26] K. WAGNER, The complexity of combinatorial problems with succinct input representation, Acta Inform.,23 (1986), pp. 325-356.
[27] ,Some observations on the connection between counting and recursion, Theoret. Comput. Sci., 47(1986), pp. 131-147.
[28] C. WRATHALL, Complete sets and the polynomial-time hierarchy, Theoret. Comput. Sci., 3 (1977),pp. 23-33.
[29] S. ZACHOS, Robustness ofprobabilistic computational complexity classes under definitional perturbations,Inform. and Control, 54 (1982), pp. 143-154.
[30] S. ZACHOS AND H. HELLER, A decisive characterization of BPP, Inform. and Control, 69 (1986),pp. 125-135.
